A test of loyalty

Abstract

We propose and test a model of loyalty in games. Players can mutually maintain loyalty by working towards a common goal that is pareto-superior to any Nash equilibrium without it. Loyalty imposes a psychological cost on defecting in an ongoing cooperation, which is thus sustained. We distinguish loyalty from reciprocity and explain how it complements guilt aversion with two dynamic games from a field experiment conducted in a Pakistani factory. The evidence supports the validity of loyalty, which has a stronger effect within than between groups.

Appendices

Appendix A: Resolution of the games with existing social preferences

1.1 A1 Reciprocity in the Trust Game

Proposition 1

For intermediate values of the reciprocity parameter, $\theta _{B,r} \in (1/50,1/40)$, there exists an equilibrium in which player A chooses SEND and player B returns with a probability $q=3-\frac{1}{20 \theta _{B,r}} \in [1/2,1]$. For the highest values of the reciprocity parameter $\theta _{B,r}\ge 1/40$, there exists an equilibrium in which A chooses SEND and player B chooses RETURN with probability one. For any value of the reciprocity parameter of player B, $\theta _{B,r}$, an equilibrium in which player A chooses to KEEP exists, sustained by the out-of-equilibrium belief that B would RETURN less than half of the time if player A were to SEND. This equilibrium does not survive forward induction when one of the other two equilibria exist.

Proof

• 1. The perceived kindness of player A by player B is equal to $\hat{\lambda }_{B}=100--20\beta - \frac{40+100--20\beta }{2}=30--10\beta$. The kindness of player B to player A in the second stage is either $K_{B}=-20$ (if she chooses to HOLD) or $K_{B}=20$ if she chooses to RETURN.

• 2. From the perspective of player A, choosing to SEND yields higher expected payoff than choosing to KEEP if his first belief on q, $\alpha \ge \frac{1}{2}$.

• 3. For a given value of her second-order beliefs $\beta$, player B prefers to RETURN whenever $40 \theta (30--10\beta ) \ge 20$. Thus, in any mixed strategy with $q \le 1$, it must hold that B is indifferent, $\alpha =\beta =q=3-\frac{1}{20 \theta }$

• 4. For any value of $\theta \in [\frac{1}{40},\frac{1}{50}]$, the above equality is satisfied with a value of $q \in [\frac{1}{2},1]$.

• 5. For values of $\theta < \frac{1}{50}$, it is satisfied with $q<\frac{1}{2}$. In that case, player A prefers to KEEP in the first stage.

• 6. Finally, for values of $\theta > \frac{1}{40}$, $40 \theta (30--10\beta ) \ge 20$ for all values of $\beta$, so that it is a dominant strategy for player B to RETURN regardless of $\beta$, and player A always prefers to SEND

• 7. For all values of $\theta ,$ player A prefers to KEEP whenever $\alpha < \frac{1}{2}$. However player B must update her beliefs to $\beta \ge \frac{1}{2}$ upon observing that player A chose to SEND. When $\theta _{B,r} \ge \frac{1}{50}$, it implies that she chooses to RETURN with probability higher than $\frac{1}{2}$. Therefore, the equilibrium in which player A chooses to KEEP only survives forward induction when no other equilibrium exists.

$\square$

1.2 A2: Guilt aversion

1.2.1 A2.a Trust Game

Proposition 2

For sufficiently high guilt aversion parameter of player B, $\theta _{B,g} \ge 20$, there exists an equilibrium in which player A chooses to SEND and player B always chooses RETURN. An equilibrium in which player A chooses to KEEP always exists, sustained by the out-of-equilibrium belief that B would RETURN less than half of the time if player A were to SEND. It however does not survive forward induction whenever $\theta _{B,g} \ge 20$. For intermediate values of the reciprocity parameter, $\theta _{B,g} \in (20,40)$, an equilibrium in which player A chooses to SEND and player B returns with a probability $q=\frac{20}{\theta _{B,g}} \in [1/2,1]$ coexists with the equilibrium in pure strategy. The mixed strategy equilibrium however does not survive a small perturbation in the beliefs.

Proof

• 1. If player A chooses to SEND, player B prefers to RETURN whenever $\theta _{B,g} \ge \frac{20}{\beta }$. Player A prefers to SEND whenever $20+40\alpha \ge 40 \Leftrightarrow \alpha > 1/2$.

• 2. As $\frac{20}{\beta }$ is decreasing in $\beta$, the smallest possible value of the guilt aversion parameter for B to RETURN corresponds to a second-order belief on the probability to RETURN $\beta =1$, and is given by $\theta _{B,g}>20$.

• 3. For a mixed strategy to exists, player B must be indifferent between HOLD and RETURN, $\beta =\alpha =q=\frac{20}{\theta _{B,g}}$. For this mixed strategy to be part of an equilibrium, player A must prefer to SEND. The smallest value of $\alpha =\beta =q$ such that A prefers to SEND is 1/2, so that the smallest value of $\theta _{B,g}$ such that $\frac{20}{\theta _{B,g}}\ge 1/2$ is $\theta _{B,g} \ge 40$. The highest value of $\theta _{B,g}$ such that $\frac{20}{\theta _{B,g}} < 1$ (at which point the mixed strategy degenerates to the pure strategy where B RETURNS with probability 1) is $\theta _{B,g} < 40$. This mixed strategy equilibrium is however not stable in the sense that a small tremble in those probabilities leads to best responses converging to a pure strategy equilibrium.

• 4. As was the case with reciprocity, the equilibrium in which player A chooses to KEEP does not survive forward induction if $\theta _{B,g} >20$, as player A should expect B to update her beliefs to $\beta \ge 1/2$ after observing that A chose to SEND and therefore $q=\beta =1$.

$\square$

1.2.2 A2.a ROSCA Game

Proposition 3

For any value of the guilt aversion parameters, an equilibrium in which both players chooses to SEND in the first stage exists and survives forward induction. No equilibrium with one or both players choosing to KEEP survives a small perturbation in the probabilities of the other player to SEND. When the guilt aversion parameter of a player i chosen by Nature to have the role of B, $\theta _{i,g} <20$, this player chooses to HOLD in the second stage. For $\theta _{i,g} \ge 20$ two equilibrium strategies in pure strategy coexist: to RETURN or to HOLD with probability one. For $\theta _{i,g} \in (20,40)$, an equilibrium strategy to return with a probability $q=\frac{20}{\theta _{B,g}} \in [1/2,1]$ coexists with the equilibria in pure strategy. The mixed strategy equilibrium however does not survive a small perturbation in the beliefs.

Proof

• Both players prefer to SEND, even if they expect to RETURN with probability one and expect the other to HOLD with probability one.

• Each player with a guilt aversion parameter $\theta _{i,g} \ge 20$ chooses to RETURN with probability 1 when selected by Nature to have the role of B if her second-order belief about $q_i$ is at least $\beta _{ij} \ge \frac{20}{\theta _{ij}}$.

• Hence, for all value of $\theta _{i,g}>20$, there exists an equilibrium in which player i chooses to RETURN with probability one and one where she chooses to HOLD with probability one.

• As both players prefer to SEND regardless of $q_1,q_2$, the equilibrium in mixed strategy exists even for values of $q_i=\frac{20}{\theta _{ij}}<1/2$. As for the TG, it is however not stable in the sense that a small tremble in those probabilities leads to best response converging to a pure strategy equilibrium.

• There is also no need for any player to update their beliefs upon observing the other choosing to SEND, as this strategy is compatible with any value of $\alpha$. Hence, if the prior belief is that the other player will HOLD, the updated belief after observing (SEND,SEND) can remain $\beta _{ij}=0$.

$\square$

Comparing the TG and the RG with guilt aversion, an equilibrium in which player B (or the player chosen by Nature to take the role of player B) chooses to RETURN with probability one when $\theta _{B,g}>20$ exists. There are two differences however. First, we should observe subjects to SEND more often in the RG. Second, we should observe subjects to RETURN more often in the TG, because after observing the other player has chosen SEND, player B updates her beliefs to $\beta \ge 1/2$, for which the only possible equilibrium value surviving a small perturbation in the beliefs is $\beta =1$, and prefers to SEND whenever $\theta _{B,g}>20$, while in the RG beliefs are not updated and choosing to HOLD is an equilibrium strategy even when $\theta _{B,g}>20$.

Appendix B. Resolution of the games with Loyalty

Proposition 4

The Trust Game with loyalty and guilt aversion displays the same equilibria as the TG with guilt aversion only.

Proof

By definition of loyalty, a single player moves before the last stage, so that no history of perceived cooperation can be built. The game is thus identical with and without loyalty. $\square$

Proposition 5

For any value of the guilt aversion parameters, an equilibrium in which both players chooses to SEND in the first stage exists and survives forward induction. No equilibrium with one or both players choosing to KEEP survives a small perturbation in the probability of the other player to SEND. There exists an equilibrium in which both players RETURN with probability one whenever $\theta _{i,g}+\theta _{i,l} \ge 20$ for both $i \in \{1,2\}$. If this conditon does not hold, similar to the case with guilt aversion only, there exists an equilibrium in which player i RETURN with probability one regardless of what player j does whenever $\theta _{i,g} \ge 20$. The other equilibria identified in Proposition 3 continue to exist.

Proof

• Both players prefer to SEND, even if they expect to RETURN with probability one and expect the other to HOLD with probability one.

• Each player with a guilt aversion parameter $\theta _{i,g}$ and loyalty parameter $\theta _{i,l}$ chooses to RETURN with probability 1 when selected by Nature to have the role of B if her first-order belief about $q_j$, $\alpha _{ij}$ and her second-order belief about $q_i$, $\beta _{ij}$ are such that $\theta _{i,g} \beta _{ij} + \theta _{i,l} \alpha _{ij} \ge 20$.

• Hence, for all value of $\theta _{i,g}+ \theta _{i,l} \ge 20$ and $\theta _{j,g}+ \theta _{j,l} \ge 20$, $j \ne i$, there exists an equilibrium in which $\alpha _{ij}=\beta _{ij}=q_1=q_2=1$, both players choose to RETURN with probability one when selected by Nature to take the role of player B.

• When, for one of the two players, $\theta _{i,g} + \theta _{i,l} < 20$, but for the other $\theta _{j,g} \ge 20$, an equilibrium in which player i chooses to HOLD and player j to RETURN when selected by Nature exists.

• If for both players $\theta _{j,g} < 20,$ only the equilibrium in which both players SEND and HOLD with probability one exists.

• There is no need for any player to update their beliefs upon observing the other choosing to SEND, as this strategy is compatible with any value of $\alpha _{ij}, \beta _{ij}$.

• As for guilt aversion, a mixed strategy would not be stable in the sense that a small tremble in $\alpha$ and $\beta$ leads to best response converging to a pure strategy equilibrium.

$\square$

Comparing the Trust Game and the ROSCA game, there are two main differences. First, in the TG player B updates her beliefs on $\beta$ upon observing a first move to SEND, so that $\beta \ge 1/2$, while this is not the case in the RG. For that reason, we should expect more RETURN in the TG. Second, the condition for an equilibrium in which players choose to RETURN to exist is less restrictive in the RG, because of the additional loyalty parameter. Asymmetric equilibria in which only one of the player RETURN exist under similar conditions as in the TG, but the condition for the symmetric equilibrium in which both RETURN is weaker in RG. Hence, depending on whether the first effect (updated beliefs) or the second (weaker existence condition) dominates, we should expect more or less RETURN in the TG than in the RG. As we are looking for indications that loyalty is an important feature of psychological utility, we can make the prediction that if subjects RETURN more in RG than in TG, the existence of loyalty is more important than updated beliefs.

Appendix C. Behavior and trust attitudes

Using logit regressions with standard errors clustered at the subject level, we explore the relationship between behavior in both games and the self-reported measures of perceptions regarding the trust attitudes of others. We run regressions separately for interactions within group and across the factory on SEND (= 1 if SEND, = 0 if KEEP) for player A and RETURN (= 1 if RETURN, = 0 if HOLD) for player B with the independent variables RG (= 1 if RG, = 0 for TG), as well as trust attitudes with respects to others in the group, where HELPFUL-in, TRUSTWORTHY-in, RECIPROCITY-in, and FAIR-in (= 0 to 1), and factory, where HELPFUL-out, TRUSTWORTHY-out, RECIPROCITY-out, and FAIR-out (= 0 to 1). We control for SEX (= 1 if male, = 0 if female) and AGE (= 15-58).

Models 1 and 2 show with the positive and significant RG estimates that SEND and RETURN are higher in RG than TG when played within the group. Further, model 2 shows that RETURN increases with the perceived helpfulness of others (HELPFUL-in is positive and significant). The positive sign of RG in models 5 and 6 corroborate that subjects also SEND and RETURN more in RG than TG in outgroup interactions, and it is significant for SEND but not for RETURN, which is consistent with the weaker effect of outgroup loyalty in the RG (H2) and social proximity in the TG (H3).

Further regressions interact RG with the trust attitudes in models 3, 4, 7, and 8. We find that reciprocity increases with self-perceived trustworthiness (Model 6) but decreases with the perceived helpfulness of the outgroup (models 5 and 7) in contrast to how it increases for the ingroup (model 2). This suggests a tendency to reciprocate the ingroup but take advantage of the outgroup despite their positive intentions, which implies group-specific preference parameter values, despite no significant difference in self-reported RECIPROCITY to the ingroup and outgroup. Otherwise, the differences between effects of trust attitudes on behavior in RG and TG are not significant.